Superadiabatic small-scale combustor with counter-flow heat exchange: Flame structure and limits to narrow-channel approximation
Introduction
Recent challenges for combustion science linked to the reduction of harmful emissions and the increase of efficiency require the development and validation of new combustion applications that use low-energy containing mixtures. Among them we find small-scale devices designed for hydrogen production and based on reforming rich fuel–air mixtures by means of partial oxidation. Another class of such devices consists in small burners for low-energy fuel-residual gases to provide, in particular, energy supply for portable electronic devices. In both cases, the direct chemical conversion per se is hardly viable due to low heat output and high heat losses. Comprehensive reviews regarding these micro combustion devices can be found in [1], [2], [3], [4], [5], [6].
A viable solution for combustion of low-energy-containing mixtures might be the use of small-scale devices with heat exchanging between adjacent counter-flow streams. The idea of this sort of devices should be traced back to [7], [8], [9] and the corresponding studies have received significant attention over the past two decades [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].
An increase in the efficiency of heat exchange between counterflow streams can be achieved by reducing the thickness of the channels when it becomes comparable with the thermal thickness of the flame. A simple one-dimensional model for such devices was developed in [11], [12], [13] where the flame-sheet approximation for the reaction rate was also adopted. These simplifications allowed an analytical treatment both for steady state solutions and for their stability analysis. Various numerical studies based on two-dimensional governing equations were carried out in [14], [15], [16], [17] for different burner configurations. The influence of power extraction on combustion regimes has been reported recently in [18] and the production viability of hydrogen-rich syngas from methanol has been explored in [19].
The description of processes in such devices requires a significant number of parameters, to account for the physicochemical properties of combustible mixtures and for the geometrical characteristics of the devices themselves. For this reason, the use of simplified one-dimensional models for the parametric analysis has a clear advantage due to the lower computational cost of numerical simulations. Typically, these one-dimensional models are built in an ad-hoc manner, without proper and rigorous studies of their applicability. In contrast to this, we believe that proper construction of such simplified models should be based on asymptotic methods. It is also well known that asymptotic considerations provide approximations that could very well be valid beyond their strict (mathematic) limit of validity. However, the range of validity cannot be obtained within the framework of the asymptotic methods themselves. Instead, it has to be deduced from the comparison of results of the one-dimensional model to results of the complete multidimensional system.
The purpose of the current work is to justify the use of one-dimensional models for devices with heat recirculation and to determine the limits of their applicability. With this aim, the two-dimensional model taken as a starting point is investigated numerically first. After that, the limit of narrow channels is developed and the results are compared with those obtained for the complete problem. The article is arranged as follows: in Section 2, the general formulation is given; a short description of the numerical treatment is given in Section 3; the asymptotic considerations leading to the one-dimensional narrow channel approximation are presented in Section 4; the asymptotic results are compared with the direct two-dimensional numerical calculations in Section 5; in addition, the high activation energy limit is used within the narrow channel approximation in Section 6. Finally, conclusions are drawn in the last section.
Section snippets
General formulation
A schematic representation of the idealized counter-flow combustor under consideration is given in Fig. 1. The device consists of an array of planar parallel channels of height H separated from each other by walls of thickness Hw. A combustible mixture at initial temperature T0 flows in opposite directions in adjacent channels with a fixed mass flow rate, M0, in every channel. The configuration is periodic in the y-direction. We assume that the walls are thermally adiabatic except for segments
Numerical treatment
All two-dimensional computations were carried out in a finite domain, with xmax significantly larger than ℓ/2. The spatial derivatives were discretized on a uniform grid using second order, three-point central differences. The steady counterpart () of the governing equations was solved using a Gauss–Seidel method with over-relaxation. The solutions were obtained using two iterating methods. In the first method, the value of the flow rate, m, was fixed. Only solutions
Narrow-channel approximation
A convincing asymptotic reduction of the two-dimensional problem to its one-dimensional counterpart has evident numerical advantages for narrow channels. In addition to reducing numerical costs, this allows to recover properly the heat-exchange parameter which should be used at low a.
The asymptotic procedure carried out below is similar to that described in [25] where a single adiabatic channel was considered. It will not be reproduced here in all details. For the sake of brevity we reproduce
Numerical results
The main aim of the present study is to compare the solutions based on the two-dimensional formulation given by Eqs. (3)–(9) and calculated for finite values of with those obtained from Eqs. (22)–(26) derived as a → 0. It should be noted that steady-state solutions exist only for m ≥ 1 in the given burner configuration. For m < 1 the flame can enter and propagate inside the adiabatic segments.
In accordance with the results obtained previously in various studies, for example see [11], [12]
Large activation energy limit
It is interesting to compare the numerical results calculated with the spatially distributed reaction rate (finite β) and those of large activation energy approximation (β → ∞) known as the flame-sheet model. In point of fact, the mathematical formulation of the flame sheet model consists in substituting the reaction rate given by Eq. (10) with where δ( · ) is the Dirac delta function. This approximation was used in [11], [12], [13] for flames in heat-recirculating devices.
For the
Concluding discussion
Building simplified models for numerical analysis is sometimes difficult, especially when a significant number of physical processes must be included. One way to do this is to use asymptotic methods. However, after a simplified model is built, a legitimate question arises about the extent of its application. As a rule, the answer to this question cannot be obtained from the asymptotic construction itself, and comparison with the results obtained in the framework of a more general model is
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The authors acknowledge the support of Spanish MEC under project #ENE2015-65852-C2-2-R (MINECO/FEDER, EU). VK would like to express his gratitude to Dr. B. Naud (CIEMAT) for exciting discussions.
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